## Problem

Let $X$ have a CDF $F$. Find the CDF of $X^+ = \text{max}\{0,X\}$.

## Solution

$F_{X^+}(x) = P(X^+ \leq x) = P(max(0, X) \leq x)$

When $x$ is less than 0, the probability is 0, and when it is greater or equal to zero, this is just $P(X \leq x) = F(x)$

So

$F_{X^+}(x) = \begin{cases} 0 & x < 0 \\ F(x) & x \geq 0 \end{cases} $